which arc is congruent to $\widehat{eh}$?

$\bigcirc$ $\widehat{gh}$
$\bigcirc$ $\widehat{fh}$
$\bigcirc$ $\widehat{ge}$
$\bigcirc$ $\widehat{fg}$

Question

which arc is congruent to $\widehat{eh}$?

$\bigcirc$ $\widehat{gh}$
$\bigcirc$ $\widehat{fh}$
$\bigcirc$ $\widehat{ge}$
$\bigcirc$ $\widehat{fg}$

Accepted Answer

{ "quality": "clear", "question_count": 1, "skills_matched": [ "step_cot", "multiple_choice_formatter", "explain_then_answer", "quick" ], "needs_drawing": false, "concepts_used": [ "Central Angle Theorem", "Congruent Arcs", "Circle Geometry" ], "new_concepts": [], "current_concepts": [ "Central Angle Theorem", "Congruent Arcs", "Circle Geometry" ] } ## Find the measure of the central angle for arc EH $$ \begin{aligned} &FH \text{ is a line segment passing through the center } D \text{, so } \angle FDH = 180^\circ.\ &m\angle EDF = 55^\circ\ &m\angle EDH = 180^\circ - m\angle EDF = 180^\circ - 55^\circ = 125^\circ\ &m\widehat{EH} = m\angle EDH = 125^\circ \end{aligned} $$ ## Find the measures of the other central angles and arcs $$ \begin{aligned} &m\angle FDG = 70^\circ \implies m\widehat{FG} = 70^\circ\ &m\angle GDH = 110^\circ \implies m\widehat{GH} = 110^\circ\ &m\angle EDG = m\angle EDF + m\angle FDG = 55^\circ + 70^\circ = 125^\circ\ &m\widehat{GE} = m\angle EDG = 125^\circ \end{aligned} $$ ## Compare arc measures to find the congruent arc $$ \begin{aligned} &m\widehat{EH} = 125^\circ\ &m\widehat{GE} = 125^\circ\ &m\widehat{EH} = m\widehat{GE} \implies \widehat{EH} \cong \widehat{GE} \end{aligned} $$

(A) $\widehat{GH}$

(B) $\widehat{FH}$

(C) $\widehat{GE}$

(D) $\widehat{FG}$

{ "subject": "Mathematics", "question_type": "Multiple Choice", "knowledge_point": [ "Mathematics", "Geometry", "Circle Geometry" ] }

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QUESTION IMAGE

Question

Explanation:

Find the measure of the central angle for arc EH

Find the measures of the other central angles and arcs

Compare arc measures to find the congruent arc

Answer: